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Unformatted text preview: Math 135: Lecture 5: Existential Quantifiers 2 The Construction Method Definition 5.1. Two integers a and b are coprime if gcd( a,b ) = 1. Proposition 5.2 (Coprimeness and Divisibility ( CAD )) . If a , b and c Z and c | ab and gcd( a,c ) = 1, then c | b . Where is the existential quantifier? Its hidden - in the definition of divides . An integer m divides an integer n , and we write m | n , if Construct Method When proving that A implies B and B uses an existential quantifier, work as follows. Construct an object with certain properties such that something happens. Be sure that the object constructed is within the universe of discourse. The construction of the object is not adequate for a proof. Show that the object has the certain properties and that something happens. Structure of an Existential Statement CAD : If a , b and c Z and c | ab and gcd( a,c ) = 1, then c | b ....
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